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Conducting monetary policy without a nominal anchor
VINCENT REINHART Division of Monetary Affairs Board of Governors of the Federal Reserve System Washington, D.C.
Conducting Monetary Policy without a Nominal Anchor* The analysis of a simple maero model with forward-looking expectations suggests that a monetary authority can operate a stable and determinate gradualist interest rate policy only as long as concern for inflation is teamed with monitoring other indicators, supporting the policy eclecticism of the recent Monetary Policy Report to Congress. In general, an interest rate-smoothing rule cannot be based solely on relative prices or rates of return, including relative commodity prices and the slope of the yield curve, or respond solely to real magnitudes, such as the output gap. Any of these indicator variables, however, complement the use of an inflation target.
1. I n t r o d u c t i o n T h e most r e c e n t Monetary Policy Report to Congress suggests that the F e d e r a l Reserve's c o n d u c t of m o n e t a r y policy is directed: (1) to m o v e the e c o n o m y toward price stability; (2) to m o n i t o r a broad variety of financial indicators, as well as the m o n e t a r y and credit aggregates; and (3) to i m p l e m e n t policy changes in m e a s u r e d steps, or to follow a policy of "gradualism" (Federal R e s e r v e 1990). This paper, formalizing these t h r e e aspects of policy, examines rules that s m o o t h the path of a p o l i c y - d e t e r m i n e d s h o r t - t e r m interest rate according to the b e h a v i o r of a m a c r o e c o n o m i c indicator or a financial m a r k e t price. Alternative specifications of policy are e m b e d d e d in a conventional I S - L M f r a m e w o r k a u g m e n t e d b y d y n a m i c price a d j u s t m e n t and forward-looking expectations. W e establish, through a series of negative lessons, that all three aspects of policy m a k i n g - - c o n c e r n s a b o u t inflation, o t h e r indicators, and g r a d u a l i s m - - a r e necessary ingredients to a well-specified reaction rule. First, we d e m o n s t r a t e that c o n c e r n for inflation and *I would like to thank Debbie Danker, Jeff Fuhrer, Don Kohn, Dave Lindsey, George Moore, Dave Small, and Bill Whitesell for their helpful discussions; I have also benefited from the comments of three anonymous referees, one of whose suggestions substantially improved the second section. Of course, I am responsible for any remaining errors, and the views expressed are my own and do not necessarily reflect the views of the Federal Reserve Board or any other members of its staff.
Journal of Macroeconomics, Fall 1991, Vol. 13, No. 4, pp. 573-596 Copyright © 1991 by Louisiana State University Press 0164-0704/91/$1.50
573
Vincent Reinhart gradualism alone, in the form of a policy rule moving the shortterm rate in response to the current inflation rate, destabilizes the system. Next, we examine a succession of alternative strategies combining gradualism and concern for other indicators, while omitting concern about inflation. Those indicators include: • the real output gap; • the gap between the actual and the "equilibriuln" concept of the price level known as P-star, as discussed in Hallman, Porter, and Small (1989); • the relative price of a storable commodity; and, • the spread between a long-term interest rate and a shortterm rate. It develops that none of these variables is suitable as the sole indicator for setting short-term nominal interest rates. Depending on parameter values, such rules would: (1) be unstable, (2) simply accommodate the current rate of inflation, or (3) fail to give investors enough information to determine uniquely all forward-looking asset prices. All these rules, however, become well behaved once concern for inflation is reintroduced. Thus, two major themes emerge. First, policy eclecticism logically follows from the dynamic properties of a conventionally specified Keynesian model. Second, as a corollary, any proposed rule for the monetary authority must have a nominal anchor. A nominal anchor, however, is a necessary but not sufficient condition for the stability of a rule.
2. Responding to Indicators of the Economy This section outlines the simple model and examines interest rate rules that respond to indicators of the real economy: the current inflation rate, the level of the real output gap, and the spread of P-star over the actual price level. The model is streamlined to magnify monetary policy's ability to control the economy. First, aggregate demand is linearly related to the real short-term interest rate, y
where 574
=
al
-
a~(i
-
~r);
(i)
Conducting
Monetary
Policy without
a Nominal Anchor
y = real income; i = nominal short-term interest rate; ~r = expected inflation rate; and al and az are positive constants. 1 Second, the demand for real money balances depends negatively on the nominal short-term interest rate and positively on income, given in cash-balance terms by m + (v~ + vzi ) = p + y ,
(9,)
where m = the nominal money stock; p = goods-price index; and vl and v2 are positive constants. Third, dynamics enter private sector behavior by assuming that goodsprice inflation sluggishly responds to excess demand. Let D represent the time derivative, that is, D x = d x / d t , so that the inflation-determination equation is, D~r = b ( y - k);
(3)
where k is the level of capacity and b is a positive constant. This ad hoc adjustment equation may be thought of as an expectational Phillips curve where anticipated inflation is a function of past inflation. Fourth, assume away all uncertainty about behavior and, indeed, even the form of the model. Investors form their expectations rationally, basing their decisions on the actual path that the economy follows in the f u t u r e - - i n other words, we assume perfect foresight, z That last assumption has no special significance in this simple form of the model. Expectations only enter into the definition of the real interest rate used in the IS equation, while the inflationadjustment equation looks backward. This changes in the following section, which introduces long-lived assets and incorporates an important role for expectations. The model, thus far, is similar to Gag~Lower-case quantity variables represent logarithms and, where possible, time subscripts are suppressed for convenience. ~Rodriguez (1978) and Reinhart (1990) investigate the implications of the perfect foresight assumption in similar Keynesian models. 575
Vincent Reinhart
non and Henderson (1988) and fairly represents the mechanics of many conventionally specified macroeconometric models that rely on inertial adjustment in both behavior and anticipations. Since this model lacks the feature that would produce a Mundell-Tobin effect--an important role for wealth, monetary policy has no long-run consequences for real variables in this economy. With the real short-term interest rate defined as r, in the steady-state, income equals capacity so that its long-run value, r*, can be written r* = (al - k ) / a 2 .
With the appropriate substitutions, the inflation-determination equation becomes D~r = ba~(r* - r ) ,
(3')
which is one form of a Wicksellian adjustment relationship: inflation accelerates when the current real rate lies below the steady-state or "natural" rate. 3 We choose among simple policy rules to close the model, turning to the recent Monetary Report to Congress (Federal Reserve 1990) for guidance in specifying that rule. The R e p o r t contains three themes on the choice of policy rule. First, the desire to keep a broad focus to policy is reflected in the Report's (1990, 108) judgment that the Federal Open Market Committee "agreed that in implementing policy, they would need to continue to consider, in addition to the behavior of money, indicators of inflationary pressures and economic growth, as well as developments in financial and foreign exchange markets." Second, the Report (1990, 115) affirms a distaste for inflation, holding that "the Federal Reserve continued to pursue a policy aimed at containing and ultimately eliminating inflation while providing support for continued economic expansion." Lastly, the Report (1990, 115) expresses policy makers' preference for gradualism, favoring "measured steps'" rather than dramatic changes in the central bank's instruments. An Inflation Rule The first rule incorporates only two of those three concerns, narrowing the policy focus to a gradualist approach in fighting in-
3Whitesell (1990) has had empirical success in estimating such a relationship. 576
Conducting Monetary Policy without a Nominal Anchor flation. Specifically, the policy rule takes an inflation-feedback form:
Di = a(~ - Ix),
(4)
where ot is a positive constant and tx is the central bank's long-run inflation target, also taken to be constant. Combining the policy rule (4) with the inflation-determination equation, (3) produces a two-variable dynamic system in (i, ~r),
-{-
D~r
-ba2
ba2
(5)
.
b(al
-
k)
The phase diagram in Figure 1 details the dynamics of this system. According to the inflation equation, the inflation rate does not change when the real rate equals the natural rate, seen as the line with unit slope, intercepting the horizontal axis at r* in the figure. Above that line, a too-low real rate creates excess goods demand, which puts upward pressure on inflation; accordingly, below the r = r* line, deficient goods demand tends to pull inflation down. The central bank, by Equation (4), responds to inflation, so pressures to move the nominal rate cease when inflation equals its long-run goal,
r
--
r,
#
i
r, Figure 1. An Inflation Rule
577
Vincent Reinhart ~. Above the horizontal ~ = Ix line, incremental tightening raises i, while below that line, an increased willingness to ease lowers i. As the arrows suggest, an (i, ~r) pair not at the steady state sets off a cyclic path, with the stability of that cycle depending on the characteristic roots to the transition matrix. 4 Since both the trace, baz (which equals the sum of the roots), and the determinant, c~ba2 (which equals the product of the roots), are positive, this system must have two positive roots. Therefore, any cycle diverges from the steady state. A possible trajectory has been drawn in the figure, with the four points labelled A through D marking where that path crosses either steady-state loci. The laws of motion dictate that the path must be vertical when it crosses the rr = Ix line and horizontal when it crosses the r = r* line. Consider the former pair of points. At A and C, policy makers have momentarily achieved their inflation goal and do not pressure the nominal rate to move. At A, however, there is deficient demand so that inflation is falling; policy, by not moving the nominal rate, allows the real rate to rise, and only in the subsequent motion, as ~r falls sufficiently below Ix, is the nominal rate cut. For part of the transition, then, policy moves procyclically, allowing the real rate to rise in an economic downturn, as at A, or fall in an economic expansion, as at C. This simple model of a dynamic economy, with its unstable oscillatory path, mandates that, with each circuit around the steady state, the nominal rate would tend to rest at either a too-high or too-low real interest rate, relative to the level required to stabilize the system. Indeed, if we were to continue the trajectory in Figure 1 over time, the points where it crossed the ~r = Ix line (the successive A and C points) would grow further apart. The single-minded policy response to inflation will always lead to a procyclical policy when inflation is at its target value. Thus, an exclusive concern for inflation, teamed with a continuous time path for the nominal rate, leads to unstable oscillations. A rule that embodies only two of the three expressed concerns of policy makers does not stabilize this system. Next, we examine alternative indicators for policy, temporarily dropping an inflation anchor from the policy rule. In each case, we show that the rule has fundamental problems that are relieved by reentering a concern for inflation. We will not entertain the other 4Boyce and DiPrima (1977, 397-409), Begg (1982. 28-51), and Sheffrin (1983, 71-87) provide clear explanations of solving dynamic foresight models. 578
Conducting Monetary Policy without a Nominal Anchor a l t e r n a t i v e - - a b a n d o n i n g gradualism b y allowing discrete changes in the nominal rate in r e s p o n s e to u n a n t i c i p a t e d events, as gradualism is a r e c u r r i n g t h e m e in most discussions of the practice of policy (Meek 1982a, 1982b). s
An Output-Gap Rule Relieved of any c o n c e r n for inflation, the central bank m i g h t guide its i n s t r u m e n t b y the real e c o n o m y . Specifically, assume that policy moves the s h o r t - t e r m interest rate according to the c u r r e n t gap b e t w e e n actual and potential output, oi
=
-
k).
The central b a n k ' s role as a stabilizer, at the least, requires that ~, which r e p r e s e n t s t h e willingness to m o v e the s h o r t - t e r m rate in response to p e r c e i v e d excess d e m a n d , be positive, e A m o r e specific range for ~ will be d e t e r m i n e d later. E v e n w h e n the m e a s u r e m e n t p r o b l e m s i n h e r e n t in capacity utilization data, all variability in the links b e t w e e n m o n e t a r y policy and aggregate d e m a n d , and all uncertainty in forecasting private and public actions are assumed away, this rule is still p o o r l y designed. Both ~r, by the form of the inflation-determination equation, and i, by the form of the policy reaction function, are c o n t i n u o u s variables so that the real s h o r t - t e r m rate is fully continuous. Simply, since inflation is p r e d e t e r m i n e d in the short run, the m o n e t a r y authority's control of the nominal rate translates into s h o r t - t e r m
Slf the dynamic system given by (5) had one continuous variable, It, and one jump variable, i, stability and uniqueness requires that the steady state is a saddlepoint, or that the determinant of the transition matrix is negative. This can only follow ff et is negative, or that the change in the nominal rate is negatively related to inflation. As discussed in Furhrer and Moore (1989) and Reinhart (1990), some element of forward-looking behavior is required to reclaim stability. Such behavior can be embodied (a) in the policy rule by requiring a precise, stabilizing jump in the nominal rate in response ~o a shock, (b) directly in the Phillips curve by altering the backward-looking price process assumed in Equation (3), or (c) indirectly in the Phillips curve by assuming that forward-looking variables, such as financial market prices or a fiscal policy set with an eye toward an intertemporal budget restraint, enter into the aggregate demand relationship, which, in turn, substitutes into the Phillips curve. ~This rule is proposed in Hahn (1980). For a discussion of the pitfalls in responding to excess demand or other measures of the real performance of the economy, see Johnson (1989). 579
Vincent
Reinhart
control of the real rate. The policy rule may be rewritten (by subtracting Equation [3] from Equation [4]), D r = (8 - b ) a 2 ( r * - r ) ,
yielding a simple linear differential equation in the real short-term rate. 7 Clearly, the system explodes if 8 < b, or if policy makers move the pegged rate more slowly in response to excess demand than the market revalues the inflation rate in response to excess demand. But this can be made clearer if we return to our phase planes. The policy rule only holds i fixed when y equals k and r equals r*. The behavioral equation for inflation only holds "rr fixed when y equals k and r equals r*. Thus, as Figure 2 shows, the D i = 0 and D~r = 0 loci coincide when drawn in terms of i and ~r. Consider an arbitrary point that lies to the right of that line. Since the real rate is below r*, there is excess demand. Policy pressures push up the nominal interest rate and market pressures push up the inflation rate. The movement of the real interest rate--which is the rate that matters for aggregate d e m a n d - - d e p e n d s on the relative force by which these pressures are applied. 7This is the type analyzed in Boyce and DiPrima (1977, 11-22).
r,
i Figure 2. An Output-Gap Rule
580
Conducting Monetary Policy without a Nominal Anchor I f policy reacts too slowly, 8 < b, the nominal rate increases by less than inflation so that the real rate actually falls. T h e central bank i n a d v e r t e n t l y adds its own policy stimulus in r e s p o n s e to excess d e m a n d , destabilizing the e c o n o m y . I n d e e d , the 8 = 0 case r e p r e s e n t s Wicksell's (1898) cumulative inflation process, in which a p e g g e d nominal interest rate destabilizes the system after a shock. I f policy acts aggressively in response to excess d e m a n d , 8 > b, this rule can stabilize, seen in the figure as the trajectory that moves from A to the steady-state line. H o w e v e r , w h e r e the econo m y rests on that locus d e p e n d s on the shape of that trajectory. An o u t p u t rule can stabilize, b u t at the cost of loosing control of the long-run inflation rate. Policy reacts to the level of excess demand, which, b y the Phillips curve, is related to the change in inflation. T h e level of inflation p r o d u c e s no policy response so that w h e n p e r t u r b e d , the system comes to rest at w h a t e v e r xr is consistent with the starting values o f ~r and r and the behavioral parameters. Thus, initial conditions, and not an explicit policy decision, d e t e r m i n e the long-run inflation rate. 8 As shown in appendix E q u a t i o n (A2), the steady-state inflation rate will b e higher, • the smaller is 8; and • the g r e a t e r is an initial positive d e m a n d shock (or the m o r e r(0) is below r*). Graphically, the size o f 8 d e t e r m i n e s h o w quickly the trajectory bears into the steady-state loci, while the size of the initiating dist u r b a n c e d e t e r m i n e s how far the point A lies from the r = r * line. Additional c o n c e r n for inflation can a n c h o r the steady-state in~ flation rate. F o r example, r e t u r n i n g to the Monetary Report, we might write a c o m b i n e d policy as,
Di = ot(~r - IL) + 8(y - k ) , r e s p o n d i n g to inflation and a n o t h e r indicator, while holding the 8In models of the real economy, this dependence on initial conditions is known as "hysteresis." Indeed, ff the model contained a role for real wealth, the steady state for real variables would also depend on initial conditions. If such attention to current aggregate demand accurately describes Federal Reserve policy of the 1960s and 1970s, then this model helps explain the time-series properties of inflation noted by Barsky (1987). He finds that, since 1960, inflation has a unit root, or that a shock to the inflation process becomes permanently incorporated in the level of inflation. 581
Vincent Reinhart nominal rate to a continuous path. Substituting the IS e q u a t i o n to explain income and r e p e a t i n g the Phillips curve results in a twovariable system:
I D i l = r -Sad a+Sa2][:] + IS(ax-k)-e~D]. D~r L - bad bad J L b(al - k)
(7)
T h e transition matrix has a positive d e t e r m i n a n t so that the two roots to this system must have the same sign. Since the trace equals is b o t h necessary and sufficient for stability. Thus, the c o m b i n e d rule can stabilize. This can b e m a d e clear with the h e l p of F i g u r e 3. C o n c e r n s for both inflation and the o u t p u t gap separates the = 0 and D~r = 0 loci, as the f o r m e r rotates down (about the steady state d e f i n e d by the target t~). F o r comparison to the pure-inflation rule, we also include the ~r = IX line, which gives the resting pairs w h e n ~ equals zero and a is positive. As the arrows suggest, the = 0 and D-rr = 0 loci define oscillatory dynamics. C o n s i d e r a c o n v e r g e n t transition path associated with a stabilizing rule. That trajectory is det e r m i n e d by the D~r = 0 and = 0 loci and is horizontal w h e n it crosses the f o r m e r and vertical w h e n it crosses the latter. T h e
(b - ~)a~, 8 > b
Di
Di
Di
L. E3
,J/JA
.;
7( =/_z
r
r* Figure 3. A Combined Output-Gap/Inflation Rule 582
Conducting Monetary Policy without a Nominal Anchor stabilizing power of this rule can be seen at the points along that trajectory where inflation equals this goal, marked by A and B. While inflation is at its target at point A, deficient aggregate demand tends to pull it down further, which, all else equal, raises the real interest rate. In the pure-inflation rate rule of the previous section, policy would pause here and only move the nominal rate when deficient aggregate demand made itself evident in the actual inflation rate. Sufficient additional concern for the output gap means that the central bank does not pause when "n = tx but actively combats the too-high real rate by lowering the nominal rate. The trajectory slopes downward at point A (and upward at point B), showing that the policy rule correctly places countercyclical pressures on the real interest rate when inflation is at its goal but the economy is not at rest. Indeed, if we were to continue this trajectory, it would wind its way to the steady state. Importantly, there is only one choice for that steady state, given by the intersection of the Di = 0 and D~r = 0 loci. The combined rule, with sufficient concern for the output gap and some responsiveness to inflation both stabilizes and uniquely determines a steady state.
A P-star Policy Rule The search for alternative guideposts is reflected in the attention paid to the inflation indicator known as P-star in the Monetary Policy Report to Congress. As discussed in Hallman, Porter, and Small (1989), future inflation is well predicted by the gap between the current price level and that price level consistent with M2 velocity at its long-run average and output at its potential level, Pstar. In the words of the Report (1990, 116), "the historical record suggests that inflation tends to rise when actual prices are below the equilibrium level and to moderate when equilibrium prices are below actual." Within our model, define p* as the price level that would clear the money market at its steady-state values given the current nominal money stock. By Equation (2), p*=m-k+vl+v~(r*+
~).
(2')
Since velocity is interest sensitive, the long-run level of velocity depends on the steady-state nominal interest rate, unlike the Hallman, Porter, and Small specification. The deviation of actual from equilibrium prices can be written by subtracting current moneymarket clearing (2) from its long-run values (2'), 583
Vincent Reinhart
/9* - p = (y - k) + v2[(r* + Ix) - i], and depends on the current o u t p u t gap and the current nominal rate relative to its long-run value. Thus, a P-star rule combines concern for the output gap and inflation (in terms of the nominal interest rate) in one term. We now address if this one term can form the exclusive basis for policy. The natural form of the rule would be to move the nominal rate up w h e n p* indicates incipient inflation pressures, as in: (8)
D i = 8 ( p * - p) ,
where 8 is a positive constant. Substituting the definition of p*, D i = ~{(y - k) + v2[(r* + Ix) - i]},
this rule reduces to an interest-rate a d j u s t m e n t equation responding to excess demand and the level of the current relative to the steadystate nominal interest rate. The IS equation explains m o v e m e n t s in income so that (8) reduces to D i = ~{al - a 2 ( i - ~r) + v2[(r* + Ix) - i]},
which combines with the inflation-determination equation in a twovariable system:
I°iJoE
baJIlI +
~[al + v z ( r * + Ix)]1 . b(al - k )
Since the d e t e r m i n a n t of this matrix, - v 2 ~ b a 2 , is negative, this syst e m exhibits saddlepath stability, meaning that a c o n t i n u u m of (i, • r) pairs lead to the steady state. However, as both the nominal rate and the inflation rate are p r e d e t e r m i n e d , their inherited values n e e d not lie on that convergent path; if t h e y do not, the model does not contain a mechanism that moves the system t h e r e - - t h e system diverges. Thus, as was true with the inflation rule, a gradualist p* rule fails to stabilize. Meeting two of the three concerns of current policy is inadequate to the task. The simplest alternative is to include inflation within the policy rule, 584
C o n d u c t i n g M o n e t a r y Policy w i t h o u t a N o m i n a l A n c h o r Di = a(~r - bQ + 8(p* - p) ,
and capture the t h r e e concerns o f policy. It is easy to show that the resulting d y n a m i c system is stable as long as the policy parameters are chosen to meet: > ba2/(vz + a2);
and ot > 8v2 • Thus, c o n c e r n for inflation and the m o n e t a r y aggregate, indirectly c a p t u r e d in the p* - p term, permits a gradualist interest rate rule to stabilize this system.
3. Responding to Financial Market Prices
When financial market participants price long-lived assets, they must forecast r e t u r n s on c o m p e t i n g i n s t r u m e n t s o v e r the relevant holding period. T h e forward-looking n a t u r e of this process has led some analysts to suggest using r e p r e s e n t a t i v e asset prices as indicators for m o n e t a r y policy. 9 F r o m a m o d e l i n g standpoint, adding an asset requires making an assumption about h o w that asset substitutes in d e m a n d for the existing assets. T h e first section considers the indicator role of the price of a storable c o m m o d i t y that is ass u m e d to b e an i m p e r f e c t substitute for the s h o r t - t e r m g o v e r n m e n t security, while the following section considers the r e t u r n on a longterm b o n d that will b e v i e w e d as a p e r f e c t substitute for the shortterm g o v e r n m e n t security. ~° A C o m m o d i t y Price Rule To simplify matters, we consider a c o m m o d i t y d e m a n d e d only as an asset with fixed supply and no link to goods o u t p u t or the general price level (gold perhaps). T h e price of gold serves as an information variable that gives an i n d e p e n d e n t m a r k e t valuation of the inflation and interest rate outlook. Assume that the stock of this storable c o m m o d i t y is fixed at n and trades at price q.l~ F u r t h e r ,
9For a discussion, see Angell (1989) and Boughton and Branson (1988). ~°Thus, these modeling strategies contrast the portfolio-balanceapproach of Branson and Henderson (1985) and the perfect arbitrage model of Dornbusch (1976). "Continuing with our naming convention, both variables are in logarithms. 585
Vincent
Reinhart
assume that gold is an imperfect substitute for the government security so that the demand for the commodity depends positively on its own return relative to the opportunity cost of holding the safe asset, as in Barsky and Summers (1988). The return from storing the commodity is anticipated capital gains, D q , while the cost of this inventory holding is i; thus, the own rate of return on gold is D q - i, while the logarithm of the real stock of gold in terms of goods is n + q - /9. Demand takes the form n+
q -
p = v + l(Dq -
i) ,
(9)
where l and v are positive constants. Defining relative gold prices (q - p) as 0, asset market clearing may be written n+
O = v + l(DO -
r).
(9')
Hence, real capital gains depend on relative prices and the real interest rate. This asset-market relationship combines with the simple framework of the previous section consisting of an IS (Equation [1]), LM (Equation [2]), and the Phillips curve (Equation [3]). Consider the simple policy rule that revises the nominal shortterm rate according to the spread of the current over the steadystate relative commodity price, 0": D i = ¢(0 - 0 " ) .
(10)
The rule--incorporating gradualism and a concern for another ind i c a t o r - w o u l d suggest that higher relative commodity prices independently indicate that the real interest rate is lower than its steady-state value because investors want to hold more of the commodity asset. In the steady state, relative commodity prices are constant, with the long-run level of commodity prices equaling O* = v -
lr* -
n,
making the rule D i = ~(O -
v + l r * + n) .
Since both i and ~r are continuous, this equation, coupled with 586
C o n d u c t i n g M o n e t a r y Policy w i t h o u t a N o m i n a l A n c h o r
the Phillips curve, yields a real-rate-determination equation,
(10')
D r = d~(O - v + Ir* + n) - baz(r* - r) .
EOrl
[+'nv'+'+l+a'r+l
Equations (7') and (8) define a dynamic system in r and 0,
~---
DO
"4-
l/l
,
(n - v)/l
which determines two steady-state relationships between the real rate and relative commodity prices. Both of these schedules slope downward in (0, r) space and are drawn in the two panels of Figure 4. The upper panel examines the case in which the monetary policy parameter is small, 6 < baJl,
making the DO = 0 schedule flatter than the D r = 0 schedule. As the dynamic arrows suggest, this system is unstable so that no path leads to the steady state when r(0) differs from r*. Again, insufficient reaction by the central bank results in an explosive path because of the failure of i to adjust as quickly as "rr. If policy reacts more quickly, d~ > b a 2 / l ,
then the D r = 0 schedule is flatter than the DO = 0 schedule. As the dynamic arrows in the lower panel of Figure 4 suggest, for any given real short-term rate only one level of the market price of commodities sets off a dynamic path toward the steady state. That convergent path, known as the saddlepath, is labeled SS in the figure. While r is continuous, 0 is a market price that moves discretely when investors respond to the release of news. The existence of a saddlepath suggests that when an event makes r(0) differ from r*, financial markets can choose a unique level for commodity prices consistent with a stable return to the steady state. Along that path, the saddlepath, anticipated capital gains or losses satisfy the asset-market-clearing condition; fundamentals dictate the appropriate sequence of commodity prices. Along this transition path, the real rate differs from the nat587
Vincent Reinhart
@
I
o \
..................... - - - - - ~
DE
=
0
r-,
r-
r,
r
(9
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Figure 4. A Relative Commodity Price Rule ural rate so that inflation is changing. Just as with the excess-demand rule, the steady-state inflation rate depends on the length of that transition; a form of expression (A2) applies here, too. Any combination of low policy responsiveness (low ~b) and low demand sensitivity (low /) results in substantial long-lived consequences of a shock. Thus, while a relative commodity price rule can be stable, it does not determine a unique steady-state inflation rate; following only two of the three concerns of policy does not produce a satisfactory rule. 588
C o n d u c t i n g M o n e t a r y Policy w i t h o u t a N o m i n a l A n c h o r
Only the addition of a nominal anchor, a third concern, can tie down the system. The candidates for that role include the rates of goods or commodity price inflation or the rate of growth of the money stock. Any of those variables used in combination with a real commodity price target makes goods inflation in the steady state independent of initial conditions. For example, if the rule became Di = d~(0 - 0") + o~r,
then dynamics would be driven by a three-variable system in r, 7r, and 0, which is stable as long as d~ > a + ba2/l. Sufficient concern about the indicator complements concern about gradualism and inflation. Because the commodity serves as an indicator variable of no direct macroeconomic consequence, the steady state to this system is identical to the combination policy of responding to excess demand and inflation. That steady state depends on parameter values but not on the size of initiating disturbances. R e s p o n d i n g to t h e Y i e l d C u r v e
Unlike the asset just considered, we now examine the impact of a long-term asset in our model that has implications for the expenditure decision. The simplest device is to assume that the government introduces a claim that pays one unit of real purchasing power in perpetuity (as in Blanchard 1981). Let the long-term real return on this consol equal R. Expenditure, influenced by capital spending decisions with a long-time horizon, depends on the long real rate, y = al - a~R + e .
(1')
Changing expenditure behavior also changes the Wicksellian-inflation-determination equation; substituting (1') into the Phillips curve (3') results in D~r = ba2(R* - R) ;
(3")
the change in inflation now depends on the long-term real rate relative to the steady-state value, R*. Asset demands, however, depend on a shorter horizon. The expectations approach to the term structure hypothesizes that returns on comparable assets are equal over the same holding-period horizon. Maintaining this hypothesis, 589
Vincent
Reinhart i = R -
DR/$
(11)
+ ~r,
the instantaneous nominal return from a consol equals the shortterm rate of interest.~2 The arbitrage condition also takes the form, DR
= d~(R -
(11')
r),
showing that the market revises the long real rate according to the current slope of the real yield curve. Let us now consider a rule where the central bank takes advantage of the information incorporated in market pricing in setting policy: the change in the nominal short-term interest rate depends on an approximation to the slope of the nominal yield curve, x3 Di
= "y[(R + ~r) - i ] .
(12)
The similarities in (11') and (12) suggest that the policy maker is attempting to mimic the response of market rates. Control of the short-term real rate is given by the difference between the nominal short rate, determined by Equation (10), and inflation, determined by Equation (3"), D r = ~t(R -
r) -
ba~(R* -
R),
while the market revises the long-term real rate according to Equation (11), giving us two dynamic equations in the short- and longterm real rates. Figure 5 plots out the steady-state loci for this system. The D R = 0 schedule is a ray from the origin with unit slope because, by the form of the yield-curve relationship, short- and long-term real rates are equal in the long run. ~4 The D r = 0 schedule is flatter because the variation in the real short-term rate reflects partly policy decision, ~/(R - r), and partly market response to excess demand, b a 2 ( R * - R). At a point on the D r = 0 loci below R*, for example, the policy maker raises the nominal rate (because R > r) ~2This equation is an approximation to the duration-adjusted term structure (Mankiw 1986). ~3This is approximate because the true long-term nominal rate equals the longterm real rate plus long-term inflationary expectations, not ~r, the instantaneous inflation rate. More correctly, Equation (10) holds that policy makers respond to the slope of the real yield curve. ~4The presence of a liquidity premium shifts this schedule upward. 590
Conducting Monetary Policy without a Nominal Anchor DR
R
Q
=
0
Y
=
0
R*
Y
r
Figure 5. A Yield Curve Rule just as fast as the market raises inflationary expectations (because of the excess demand associated with R < R*). The intersection of the two schedules dictate the steady state, where r = R = R*. As the dynamic arrows in the figure suggest, this system responds cyclically to any shock that makes r different from R*. It can be shown that this cycle diverges from the steady state if ~/< @ and converges to the steady state if ~/ > t~. A large policy response, ensuring a convergent path to the steady state for any arbitrarily chosen (R, r) pair, presents a new problem. In the commodity-price case, a given short-term rate corresponded to a single commodity price that would lead the market to equilibrium. The joint assumptions that the market relied on fundamentals and that the system was stable implied one unique market price for commodities. No such argument can be made for the globally stable version of Figure 5. For any current r, any R that the financial market chooses generates a path that satisfies market fundamentals and stably fulfills expectations. Thus, a yield-curve rule does not tie down the market's initial choice for the long-term rate. Any arbitrary choice investors make will be confirmed by subsequent central bank action in setting shortterm rates. Ls Additionally, the steady state for nominal magnitudes tSThis uniqueness problem is associated with the work of Sargent and Wallace (1975) and McCallum (1986). 591
Vincent Reinhart depends on the initiating disturbance and the behavioral parameters. A policy stated in terms of relative yields places no special significance on the general level of yields. The long-run inflation rate depends on parameter values and the previous steady-state inflation rate. Adding a nominal anchor, or some reliance on a nominal magnitude, can render this rule determinate in two senses. First, a nominal anchor regains the uniqueness of the initial choice for the long-term rate. Stability will dictate only one possible long-term rate for each short-term rate. Second, a nominal anchor makes the steadystate inflation rate independent of initial conditions. Suppose, for instance, policy is written as
Di = ~/[(R + rr) - i] + et(-rr - ix).
(12')
The inclusion of inflation independent of the premium already appearing in nominal rates severs the automatic link between the market response, in terms of pricing R, and the policy response, in terms of setting r. It can be shown that the complete model determines real and nominal magnitudes and will be stable and unique for any nonzero ot as long as ~/ > t~; additionally, if not large enough to ensure stability by itself (~/ < t~), a nonzero ~/ lowers the value of a needed to stabilize the system.16 Thus, the presence of a nominal anchor renders the steady state unique while sufficient concern about the indicator stabilizes the system.
4. Conclusion The analysis of a simple macro model with forward-looking expectations suggests that a monetary authority can operate a stable and determinate gradualist interest-rate policy only as long as concern for inflation is teamed with monitoring other indicators, supporting the policy eclecticism of the recent Monetary Policy Report to Congress. In general, a policy rule cannot be based solely on relative prices or rates of return, including relative commodity prices and the slope of the yield curve, or respond solely to real magnitudes, such as the output gap.
IBIf policy follows a p u r e inflation rule, ~/ = 0, t h e n stability and u n i q u e n e s s requires that et > t~, that is, policy m u s t b e sufficiently c o m m i t t e d to t h e inflation target; a positive ~/ lowers this floor u n d e r a.
592
Conducting Monetary Policy without a Nominal Anchor First, a misapplied policy destabilizes the economy if it adjusts the short-term rate too sluggishly in response to a shock, regardless of the information variable guiding policy. This property, first noted by Wicksell (1898), lies at the root of academic mistrust of interest rate rules. Second, since these indicators already build in current inflationary expectations, a policy responding only to them, even if stable, lacks direction to force the economy to a particular long-run inflation rate. More specifically, the steady state depends on the transition path, so that any shock becomes incorporated into the long-run inflation rate. Third, a rule conditioned on a forward-looking asset price may ratify any potential price that the financial market chooses~ making it impossible to determine a unique equilibrium path. A yield-curve rule, in particular, has this uniqueness problem that implies that policy could set off on any of a number of paths by simply following the market's initial arbitrary choice. Any of these indicator variables, however, complement the use of an inflation target. Received: October 1989 Final version: October 1990
References Angell, Wayne D. "'Commodity Prices and Monetary Policy: Empirical and Theoretical Considerations." Paper presented at the annual meeting of the Virginia Association of Economists, March 16, 1989. Barsky, Robert B. "'The Fisher Hypothesis and the Forecastability and Persistence of Inflation.'" Journal of Monetary Economics 19 (January 1987): 3-24. Barsky, Robert B., and Lawrence H. Summers. "'Gibson's Paradox and the Gold Standard.'" Journal of Political Economy 96 (June 1988): 528-59. Begg, David K.H. The Rational Expectations Revolution in Economics: Theories and Evidence. Baltimore: Johns Hopkins Univ. Press, 1982. Blanchard, Olivier J. "Output, the Stock Market, and Interest Rates." American Economic Review 71 (March 1981): 132-43. Boughton, James D., and William H. Branson. "'Commodity Prices as Leading Indicators of Inflation." Working Paper. International Monetary Fund, October 1988. Boyce, William E., and Richard C. DiPrima. Elementary Differ593
Vincent Reinhart ential Equations and Boundary Value Problems. New York: John Wiley and Sons, 1977. Branson, William H., and Dale W. Henderson. "The Specification and Influence of Asset Markets." In Handbook of International Economics, edited by Ronald W. Jones and Peter B. Kenen, 749805. New York: North-Holland, 1985. Dornbusch, Rudiger. "Expectations and Exchange Rate Dynamics." Journal of Political Economy 84 (December 1976): 1161-76. Federal Reserve. "'Monetary Policy Report to Congress.'" Federal Reserve Bulletin 76 (March 1990): 107-19. Fuhrer, Jeffrey, and George Moore. "'The Natural Rate of Interest and Monetary Policy." Working Paper. Board of Governors of the Federal Reserve System, March 1989. Gagnon, Joseph, and Dale W. Henderson. "'Nominal Interest Rate Pegging Under Alternative Expectations Hypotheses." Working Paper. Board of Governors of the Federal Reserve System, May 1988. Hahn, Frank H. "Monetarism and Economic Theory." Economica 47 (February 1980): 1-17. Hallman, Jeffrey J., Richard D. Porter, and David H. Small. "'M2 per Unit of Potential GNP as an Anchor for the Price Level." Board of Governors of the Federal Reserve System Staff Studies 157, 1989. Johnson, Manuel H. "Issues Facing Monetary Policy in 1989." Speech before the National Association of Business Economists, Washington, D.C., February 28, 1989. Mankiw, Gregory N. "The Term Structure of Interest Rates Revisited." Brookings Papers on Economic Activity 17, no. 1 (1986): 61-96. McCallum, Bennett T. "Some Issues Concerning Interest Rate Pegging, Price Level Determinacy, and the Real Bills Doctrine." Journal of Monetary Economics 17 (January 1986): 135-60. Meek, Paul. ed. Central Bank Views on Monetary Targeting. New York: Federal Reserve Bank of New York, 1982a. Meek, Paul. U.S. Monetary Policy and Financial Markets. New York: Federal Reserve Bank of New York, 1982b. Reinhart, Vincent. "Interest Rate Smoothing and Staggered Contracting." Journal of Economics and Business 42 (February 1990): 1-16. Rodriguez, Carlos A. "'A Simple Keynesian Model of Inflation and U n e m p l o y m e n t under Rational Expectations." Weltwirtschaftliches Archiv 114, no. 1 (1978): 1-11. 594
Conducting Monetary Policy without a Nominal Anchor Sargent, Thomas J., and Neil A. Wallace. "" 'Rational' Expectations, the Optimal Monetary Instrument, and the Optimal Money Supply Rule." Journal of Political Economy 83 (April 1975): 241-54. Sheffrin, Steven M. Rational Expectations. Cambridge: Cambridge University Press, 1983. Whitesell, William C. "Inflation-Fighting with Interest Rate Rules." Working Paper. Board of Governors of the Federal Reserve System, March 1990. Wicksell, Knut. "'The Influence of the Rate of Interest on Commodity Prices." Lecture to the Economic Association in Stockholm, 1898. In Selected Papers on Economic Theory. Cambridge: Harvard University Press, 1958, 67-89.
Appendix A: The Long-Run Consequences of an Output-Gap Rule As explained in Section 2, a policy rule responding only to excess demand reduces to a single linear differential equation in the short-term real interest rate, Equation (8), the solution to which takes the form
r(t) - r* = Jr(0) - r * ] . e x p [ - ( ~ - b)aut]. When the rule is stable (~ > b), the impact of a shock that makes the current short-term rate, r(O), differ from the natural rate, r*, decays exponentially. During that adjustment, however, the inflation rate varies in response to the difference in expenditure from potential. Indeed, the change in the inflation rate up to period t consists of the cumulative impact of excess demand, t
•r(t) = at(0) + b f0 [y(s) - k i d s , or, remembering that excess demand is written as the difference between the real and natural rates, ~(t) = ~(0) - ba2[r(O) - r*]
exp[-(8 - b)a2s]ds.
Performing this integration, rr(t) = ,r(0) + [r(0) - r*]b. {exp[-(8 - b)a2t] - 1}/(8 - b ) .
(A1) 595
Vincent Reinhart N o t e t h a t as t ---> ~, ~r(t) = rr(0) - [r(0) - r*]" thus, initial c o n d i t i o n s , rr(0) a n d state inflation.
b/(8 - b) ;
(A2)
r(O), figure i m p o r t a n t l y in s t e a d y -
Appendix B: List of Variables y = k = i = ~r = r = m = /9 = p* = n q 0 R
596
= = = =
rea] i n c o m e ; potential output; n o m i n a l s h o r t - t e r m i n t e r e s t rate; e x p e c t e d inflation rate; e x p e c t e d real s h o r t - t e r m i n t e r e s t rate; n o m i n a l m o n e y stock; g o o d s - p r i c e index; p r i c e level t h a t clears t h e m o n e y m a r k e t at its s t e a d y - s t a t e values g i v e n t h e c u r r e n t n o m i n a l m o n e y stock; stock o f t h e s t o r a b l e c o m m o d i t y ; nominal price of the storable commodity; relative c o m m o d i t y prices; l o n g - t e r m real r e t u r n on t h e consol.
Conducting Monetary Policy without a Nominal Anchor* The analysis of a simple maero model with forward-looking expectations suggests that a monetary authority can operate a stable and determinate gradualist interest rate policy only as long as concern for inflation is teamed with monitoring other indicators, supporting the policy eclecticism of the recent Monetary Policy Report to Congress. In general, an interest rate-smoothing rule cannot be based solely on relative prices or rates of return, including relative commodity prices and the slope of the yield curve, or respond solely to real magnitudes, such as the output gap. Any of these indicator variables, however, complement the use of an inflation target.
1. I n t r o d u c t i o n T h e most r e c e n t Monetary Policy Report to Congress suggests that the F e d e r a l Reserve's c o n d u c t of m o n e t a r y policy is directed: (1) to m o v e the e c o n o m y toward price stability; (2) to m o n i t o r a broad variety of financial indicators, as well as the m o n e t a r y and credit aggregates; and (3) to i m p l e m e n t policy changes in m e a s u r e d steps, or to follow a policy of "gradualism" (Federal R e s e r v e 1990). This paper, formalizing these t h r e e aspects of policy, examines rules that s m o o t h the path of a p o l i c y - d e t e r m i n e d s h o r t - t e r m interest rate according to the b e h a v i o r of a m a c r o e c o n o m i c indicator or a financial m a r k e t price. Alternative specifications of policy are e m b e d d e d in a conventional I S - L M f r a m e w o r k a u g m e n t e d b y d y n a m i c price a d j u s t m e n t and forward-looking expectations. W e establish, through a series of negative lessons, that all three aspects of policy m a k i n g - - c o n c e r n s a b o u t inflation, o t h e r indicators, and g r a d u a l i s m - - a r e necessary ingredients to a well-specified reaction rule. First, we d e m o n s t r a t e that c o n c e r n for inflation and *I would like to thank Debbie Danker, Jeff Fuhrer, Don Kohn, Dave Lindsey, George Moore, Dave Small, and Bill Whitesell for their helpful discussions; I have also benefited from the comments of three anonymous referees, one of whose suggestions substantially improved the second section. Of course, I am responsible for any remaining errors, and the views expressed are my own and do not necessarily reflect the views of the Federal Reserve Board or any other members of its staff.
Journal of Macroeconomics, Fall 1991, Vol. 13, No. 4, pp. 573-596 Copyright © 1991 by Louisiana State University Press 0164-0704/91/$1.50
573
Vincent Reinhart gradualism alone, in the form of a policy rule moving the shortterm rate in response to the current inflation rate, destabilizes the system. Next, we examine a succession of alternative strategies combining gradualism and concern for other indicators, while omitting concern about inflation. Those indicators include: • the real output gap; • the gap between the actual and the "equilibriuln" concept of the price level known as P-star, as discussed in Hallman, Porter, and Small (1989); • the relative price of a storable commodity; and, • the spread between a long-term interest rate and a shortterm rate. It develops that none of these variables is suitable as the sole indicator for setting short-term nominal interest rates. Depending on parameter values, such rules would: (1) be unstable, (2) simply accommodate the current rate of inflation, or (3) fail to give investors enough information to determine uniquely all forward-looking asset prices. All these rules, however, become well behaved once concern for inflation is reintroduced. Thus, two major themes emerge. First, policy eclecticism logically follows from the dynamic properties of a conventionally specified Keynesian model. Second, as a corollary, any proposed rule for the monetary authority must have a nominal anchor. A nominal anchor, however, is a necessary but not sufficient condition for the stability of a rule.
2. Responding to Indicators of the Economy This section outlines the simple model and examines interest rate rules that respond to indicators of the real economy: the current inflation rate, the level of the real output gap, and the spread of P-star over the actual price level. The model is streamlined to magnify monetary policy's ability to control the economy. First, aggregate demand is linearly related to the real short-term interest rate, y
where 574
=
al
-
a~(i
-
~r);
(i)
Conducting
Monetary
Policy without
a Nominal Anchor
y = real income; i = nominal short-term interest rate; ~r = expected inflation rate; and al and az are positive constants. 1 Second, the demand for real money balances depends negatively on the nominal short-term interest rate and positively on income, given in cash-balance terms by m + (v~ + vzi ) = p + y ,
(9,)
where m = the nominal money stock; p = goods-price index; and vl and v2 are positive constants. Third, dynamics enter private sector behavior by assuming that goodsprice inflation sluggishly responds to excess demand. Let D represent the time derivative, that is, D x = d x / d t , so that the inflation-determination equation is, D~r = b ( y - k);
(3)
where k is the level of capacity and b is a positive constant. This ad hoc adjustment equation may be thought of as an expectational Phillips curve where anticipated inflation is a function of past inflation. Fourth, assume away all uncertainty about behavior and, indeed, even the form of the model. Investors form their expectations rationally, basing their decisions on the actual path that the economy follows in the f u t u r e - - i n other words, we assume perfect foresight, z That last assumption has no special significance in this simple form of the model. Expectations only enter into the definition of the real interest rate used in the IS equation, while the inflationadjustment equation looks backward. This changes in the following section, which introduces long-lived assets and incorporates an important role for expectations. The model, thus far, is similar to Gag~Lower-case quantity variables represent logarithms and, where possible, time subscripts are suppressed for convenience. ~Rodriguez (1978) and Reinhart (1990) investigate the implications of the perfect foresight assumption in similar Keynesian models. 575
Vincent Reinhart
non and Henderson (1988) and fairly represents the mechanics of many conventionally specified macroeconometric models that rely on inertial adjustment in both behavior and anticipations. Since this model lacks the feature that would produce a Mundell-Tobin effect--an important role for wealth, monetary policy has no long-run consequences for real variables in this economy. With the real short-term interest rate defined as r, in the steady-state, income equals capacity so that its long-run value, r*, can be written r* = (al - k ) / a 2 .
With the appropriate substitutions, the inflation-determination equation becomes D~r = ba~(r* - r ) ,
(3')
which is one form of a Wicksellian adjustment relationship: inflation accelerates when the current real rate lies below the steady-state or "natural" rate. 3 We choose among simple policy rules to close the model, turning to the recent Monetary Report to Congress (Federal Reserve 1990) for guidance in specifying that rule. The R e p o r t contains three themes on the choice of policy rule. First, the desire to keep a broad focus to policy is reflected in the Report's (1990, 108) judgment that the Federal Open Market Committee "agreed that in implementing policy, they would need to continue to consider, in addition to the behavior of money, indicators of inflationary pressures and economic growth, as well as developments in financial and foreign exchange markets." Second, the Report (1990, 115) affirms a distaste for inflation, holding that "the Federal Reserve continued to pursue a policy aimed at containing and ultimately eliminating inflation while providing support for continued economic expansion." Lastly, the Report (1990, 115) expresses policy makers' preference for gradualism, favoring "measured steps'" rather than dramatic changes in the central bank's instruments. An Inflation Rule The first rule incorporates only two of those three concerns, narrowing the policy focus to a gradualist approach in fighting in-
3Whitesell (1990) has had empirical success in estimating such a relationship. 576
Conducting Monetary Policy without a Nominal Anchor flation. Specifically, the policy rule takes an inflation-feedback form:
Di = a(~ - Ix),
(4)
where ot is a positive constant and tx is the central bank's long-run inflation target, also taken to be constant. Combining the policy rule (4) with the inflation-determination equation, (3) produces a two-variable dynamic system in (i, ~r),
-{-
D~r
-ba2
ba2
(5)
.
b(al
-
k)
The phase diagram in Figure 1 details the dynamics of this system. According to the inflation equation, the inflation rate does not change when the real rate equals the natural rate, seen as the line with unit slope, intercepting the horizontal axis at r* in the figure. Above that line, a too-low real rate creates excess goods demand, which puts upward pressure on inflation; accordingly, below the r = r* line, deficient goods demand tends to pull inflation down. The central bank, by Equation (4), responds to inflation, so pressures to move the nominal rate cease when inflation equals its long-run goal,
r
--
r,
#
i
r, Figure 1. An Inflation Rule
577
Vincent Reinhart ~. Above the horizontal ~ = Ix line, incremental tightening raises i, while below that line, an increased willingness to ease lowers i. As the arrows suggest, an (i, ~r) pair not at the steady state sets off a cyclic path, with the stability of that cycle depending on the characteristic roots to the transition matrix. 4 Since both the trace, baz (which equals the sum of the roots), and the determinant, c~ba2 (which equals the product of the roots), are positive, this system must have two positive roots. Therefore, any cycle diverges from the steady state. A possible trajectory has been drawn in the figure, with the four points labelled A through D marking where that path crosses either steady-state loci. The laws of motion dictate that the path must be vertical when it crosses the rr = Ix line and horizontal when it crosses the r = r* line. Consider the former pair of points. At A and C, policy makers have momentarily achieved their inflation goal and do not pressure the nominal rate to move. At A, however, there is deficient demand so that inflation is falling; policy, by not moving the nominal rate, allows the real rate to rise, and only in the subsequent motion, as ~r falls sufficiently below Ix, is the nominal rate cut. For part of the transition, then, policy moves procyclically, allowing the real rate to rise in an economic downturn, as at A, or fall in an economic expansion, as at C. This simple model of a dynamic economy, with its unstable oscillatory path, mandates that, with each circuit around the steady state, the nominal rate would tend to rest at either a too-high or too-low real interest rate, relative to the level required to stabilize the system. Indeed, if we were to continue the trajectory in Figure 1 over time, the points where it crossed the ~r = Ix line (the successive A and C points) would grow further apart. The single-minded policy response to inflation will always lead to a procyclical policy when inflation is at its target value. Thus, an exclusive concern for inflation, teamed with a continuous time path for the nominal rate, leads to unstable oscillations. A rule that embodies only two of the three expressed concerns of policy makers does not stabilize this system. Next, we examine alternative indicators for policy, temporarily dropping an inflation anchor from the policy rule. In each case, we show that the rule has fundamental problems that are relieved by reentering a concern for inflation. We will not entertain the other 4Boyce and DiPrima (1977, 397-409), Begg (1982. 28-51), and Sheffrin (1983, 71-87) provide clear explanations of solving dynamic foresight models. 578
Conducting Monetary Policy without a Nominal Anchor a l t e r n a t i v e - - a b a n d o n i n g gradualism b y allowing discrete changes in the nominal rate in r e s p o n s e to u n a n t i c i p a t e d events, as gradualism is a r e c u r r i n g t h e m e in most discussions of the practice of policy (Meek 1982a, 1982b). s
An Output-Gap Rule Relieved of any c o n c e r n for inflation, the central bank m i g h t guide its i n s t r u m e n t b y the real e c o n o m y . Specifically, assume that policy moves the s h o r t - t e r m interest rate according to the c u r r e n t gap b e t w e e n actual and potential output, oi
=
-
k).
The central b a n k ' s role as a stabilizer, at the least, requires that ~, which r e p r e s e n t s t h e willingness to m o v e the s h o r t - t e r m rate in response to p e r c e i v e d excess d e m a n d , be positive, e A m o r e specific range for ~ will be d e t e r m i n e d later. E v e n w h e n the m e a s u r e m e n t p r o b l e m s i n h e r e n t in capacity utilization data, all variability in the links b e t w e e n m o n e t a r y policy and aggregate d e m a n d , and all uncertainty in forecasting private and public actions are assumed away, this rule is still p o o r l y designed. Both ~r, by the form of the inflation-determination equation, and i, by the form of the policy reaction function, are c o n t i n u o u s variables so that the real s h o r t - t e r m rate is fully continuous. Simply, since inflation is p r e d e t e r m i n e d in the short run, the m o n e t a r y authority's control of the nominal rate translates into s h o r t - t e r m
Slf the dynamic system given by (5) had one continuous variable, It, and one jump variable, i, stability and uniqueness requires that the steady state is a saddlepoint, or that the determinant of the transition matrix is negative. This can only follow ff et is negative, or that the change in the nominal rate is negatively related to inflation. As discussed in Furhrer and Moore (1989) and Reinhart (1990), some element of forward-looking behavior is required to reclaim stability. Such behavior can be embodied (a) in the policy rule by requiring a precise, stabilizing jump in the nominal rate in response ~o a shock, (b) directly in the Phillips curve by altering the backward-looking price process assumed in Equation (3), or (c) indirectly in the Phillips curve by assuming that forward-looking variables, such as financial market prices or a fiscal policy set with an eye toward an intertemporal budget restraint, enter into the aggregate demand relationship, which, in turn, substitutes into the Phillips curve. ~This rule is proposed in Hahn (1980). For a discussion of the pitfalls in responding to excess demand or other measures of the real performance of the economy, see Johnson (1989). 579
Vincent
Reinhart
control of the real rate. The policy rule may be rewritten (by subtracting Equation [3] from Equation [4]), D r = (8 - b ) a 2 ( r * - r ) ,
yielding a simple linear differential equation in the real short-term rate. 7 Clearly, the system explodes if 8 < b, or if policy makers move the pegged rate more slowly in response to excess demand than the market revalues the inflation rate in response to excess demand. But this can be made clearer if we return to our phase planes. The policy rule only holds i fixed when y equals k and r equals r*. The behavioral equation for inflation only holds "rr fixed when y equals k and r equals r*. Thus, as Figure 2 shows, the D i = 0 and D~r = 0 loci coincide when drawn in terms of i and ~r. Consider an arbitrary point that lies to the right of that line. Since the real rate is below r*, there is excess demand. Policy pressures push up the nominal interest rate and market pressures push up the inflation rate. The movement of the real interest rate--which is the rate that matters for aggregate d e m a n d - - d e p e n d s on the relative force by which these pressures are applied. 7This is the type analyzed in Boyce and DiPrima (1977, 11-22).
r,
i Figure 2. An Output-Gap Rule
580
Conducting Monetary Policy without a Nominal Anchor I f policy reacts too slowly, 8 < b, the nominal rate increases by less than inflation so that the real rate actually falls. T h e central bank i n a d v e r t e n t l y adds its own policy stimulus in r e s p o n s e to excess d e m a n d , destabilizing the e c o n o m y . I n d e e d , the 8 = 0 case r e p r e s e n t s Wicksell's (1898) cumulative inflation process, in which a p e g g e d nominal interest rate destabilizes the system after a shock. I f policy acts aggressively in response to excess d e m a n d , 8 > b, this rule can stabilize, seen in the figure as the trajectory that moves from A to the steady-state line. H o w e v e r , w h e r e the econo m y rests on that locus d e p e n d s on the shape of that trajectory. An o u t p u t rule can stabilize, b u t at the cost of loosing control of the long-run inflation rate. Policy reacts to the level of excess demand, which, b y the Phillips curve, is related to the change in inflation. T h e level of inflation p r o d u c e s no policy response so that w h e n p e r t u r b e d , the system comes to rest at w h a t e v e r xr is consistent with the starting values o f ~r and r and the behavioral parameters. Thus, initial conditions, and not an explicit policy decision, d e t e r m i n e the long-run inflation rate. 8 As shown in appendix E q u a t i o n (A2), the steady-state inflation rate will b e higher, • the smaller is 8; and • the g r e a t e r is an initial positive d e m a n d shock (or the m o r e r(0) is below r*). Graphically, the size o f 8 d e t e r m i n e s h o w quickly the trajectory bears into the steady-state loci, while the size of the initiating dist u r b a n c e d e t e r m i n e s how far the point A lies from the r = r * line. Additional c o n c e r n for inflation can a n c h o r the steady-state in~ flation rate. F o r example, r e t u r n i n g to the Monetary Report, we might write a c o m b i n e d policy as,
Di = ot(~r - IL) + 8(y - k ) , r e s p o n d i n g to inflation and a n o t h e r indicator, while holding the 8In models of the real economy, this dependence on initial conditions is known as "hysteresis." Indeed, ff the model contained a role for real wealth, the steady state for real variables would also depend on initial conditions. If such attention to current aggregate demand accurately describes Federal Reserve policy of the 1960s and 1970s, then this model helps explain the time-series properties of inflation noted by Barsky (1987). He finds that, since 1960, inflation has a unit root, or that a shock to the inflation process becomes permanently incorporated in the level of inflation. 581
Vincent Reinhart nominal rate to a continuous path. Substituting the IS e q u a t i o n to explain income and r e p e a t i n g the Phillips curve results in a twovariable system:
I D i l = r -Sad a+Sa2][:] + IS(ax-k)-e~D]. D~r L - bad bad J L b(al - k)
(7)
T h e transition matrix has a positive d e t e r m i n a n t so that the two roots to this system must have the same sign. Since the trace equals is b o t h necessary and sufficient for stability. Thus, the c o m b i n e d rule can stabilize. This can b e m a d e clear with the h e l p of F i g u r e 3. C o n c e r n s for both inflation and the o u t p u t gap separates the = 0 and D~r = 0 loci, as the f o r m e r rotates down (about the steady state d e f i n e d by the target t~). F o r comparison to the pure-inflation rule, we also include the ~r = IX line, which gives the resting pairs w h e n ~ equals zero and a is positive. As the arrows suggest, the = 0 and D-rr = 0 loci define oscillatory dynamics. C o n s i d e r a c o n v e r g e n t transition path associated with a stabilizing rule. That trajectory is det e r m i n e d by the D~r = 0 and = 0 loci and is horizontal w h e n it crosses the f o r m e r and vertical w h e n it crosses the latter. T h e
(b - ~)a~, 8 > b
Di
Di
Di
L. E3
,J/JA
.;
7( =/_z
r
r* Figure 3. A Combined Output-Gap/Inflation Rule 582
Conducting Monetary Policy without a Nominal Anchor stabilizing power of this rule can be seen at the points along that trajectory where inflation equals this goal, marked by A and B. While inflation is at its target at point A, deficient aggregate demand tends to pull it down further, which, all else equal, raises the real interest rate. In the pure-inflation rate rule of the previous section, policy would pause here and only move the nominal rate when deficient aggregate demand made itself evident in the actual inflation rate. Sufficient additional concern for the output gap means that the central bank does not pause when "n = tx but actively combats the too-high real rate by lowering the nominal rate. The trajectory slopes downward at point A (and upward at point B), showing that the policy rule correctly places countercyclical pressures on the real interest rate when inflation is at its goal but the economy is not at rest. Indeed, if we were to continue this trajectory, it would wind its way to the steady state. Importantly, there is only one choice for that steady state, given by the intersection of the Di = 0 and D~r = 0 loci. The combined rule, with sufficient concern for the output gap and some responsiveness to inflation both stabilizes and uniquely determines a steady state.
A P-star Policy Rule The search for alternative guideposts is reflected in the attention paid to the inflation indicator known as P-star in the Monetary Policy Report to Congress. As discussed in Hallman, Porter, and Small (1989), future inflation is well predicted by the gap between the current price level and that price level consistent with M2 velocity at its long-run average and output at its potential level, Pstar. In the words of the Report (1990, 116), "the historical record suggests that inflation tends to rise when actual prices are below the equilibrium level and to moderate when equilibrium prices are below actual." Within our model, define p* as the price level that would clear the money market at its steady-state values given the current nominal money stock. By Equation (2), p*=m-k+vl+v~(r*+
~).
(2')
Since velocity is interest sensitive, the long-run level of velocity depends on the steady-state nominal interest rate, unlike the Hallman, Porter, and Small specification. The deviation of actual from equilibrium prices can be written by subtracting current moneymarket clearing (2) from its long-run values (2'), 583
Vincent Reinhart
/9* - p = (y - k) + v2[(r* + Ix) - i], and depends on the current o u t p u t gap and the current nominal rate relative to its long-run value. Thus, a P-star rule combines concern for the output gap and inflation (in terms of the nominal interest rate) in one term. We now address if this one term can form the exclusive basis for policy. The natural form of the rule would be to move the nominal rate up w h e n p* indicates incipient inflation pressures, as in: (8)
D i = 8 ( p * - p) ,
where 8 is a positive constant. Substituting the definition of p*, D i = ~{(y - k) + v2[(r* + Ix) - i]},
this rule reduces to an interest-rate a d j u s t m e n t equation responding to excess demand and the level of the current relative to the steadystate nominal interest rate. The IS equation explains m o v e m e n t s in income so that (8) reduces to D i = ~{al - a 2 ( i - ~r) + v2[(r* + Ix) - i]},
which combines with the inflation-determination equation in a twovariable system:
I°iJoE
baJIlI +
~[al + v z ( r * + Ix)]1 . b(al - k )
Since the d e t e r m i n a n t of this matrix, - v 2 ~ b a 2 , is negative, this syst e m exhibits saddlepath stability, meaning that a c o n t i n u u m of (i, • r) pairs lead to the steady state. However, as both the nominal rate and the inflation rate are p r e d e t e r m i n e d , their inherited values n e e d not lie on that convergent path; if t h e y do not, the model does not contain a mechanism that moves the system t h e r e - - t h e system diverges. Thus, as was true with the inflation rule, a gradualist p* rule fails to stabilize. Meeting two of the three concerns of current policy is inadequate to the task. The simplest alternative is to include inflation within the policy rule, 584
C o n d u c t i n g M o n e t a r y Policy w i t h o u t a N o m i n a l A n c h o r Di = a(~r - bQ + 8(p* - p) ,
and capture the t h r e e concerns o f policy. It is easy to show that the resulting d y n a m i c system is stable as long as the policy parameters are chosen to meet: > ba2/(vz + a2);
and ot > 8v2 • Thus, c o n c e r n for inflation and the m o n e t a r y aggregate, indirectly c a p t u r e d in the p* - p term, permits a gradualist interest rate rule to stabilize this system.
3. Responding to Financial Market Prices
When financial market participants price long-lived assets, they must forecast r e t u r n s on c o m p e t i n g i n s t r u m e n t s o v e r the relevant holding period. T h e forward-looking n a t u r e of this process has led some analysts to suggest using r e p r e s e n t a t i v e asset prices as indicators for m o n e t a r y policy. 9 F r o m a m o d e l i n g standpoint, adding an asset requires making an assumption about h o w that asset substitutes in d e m a n d for the existing assets. T h e first section considers the indicator role of the price of a storable c o m m o d i t y that is ass u m e d to b e an i m p e r f e c t substitute for the s h o r t - t e r m g o v e r n m e n t security, while the following section considers the r e t u r n on a longterm b o n d that will b e v i e w e d as a p e r f e c t substitute for the shortterm g o v e r n m e n t security. ~° A C o m m o d i t y Price Rule To simplify matters, we consider a c o m m o d i t y d e m a n d e d only as an asset with fixed supply and no link to goods o u t p u t or the general price level (gold perhaps). T h e price of gold serves as an information variable that gives an i n d e p e n d e n t m a r k e t valuation of the inflation and interest rate outlook. Assume that the stock of this storable c o m m o d i t y is fixed at n and trades at price q.l~ F u r t h e r ,
9For a discussion, see Angell (1989) and Boughton and Branson (1988). ~°Thus, these modeling strategies contrast the portfolio-balanceapproach of Branson and Henderson (1985) and the perfect arbitrage model of Dornbusch (1976). "Continuing with our naming convention, both variables are in logarithms. 585
Vincent
Reinhart
assume that gold is an imperfect substitute for the government security so that the demand for the commodity depends positively on its own return relative to the opportunity cost of holding the safe asset, as in Barsky and Summers (1988). The return from storing the commodity is anticipated capital gains, D q , while the cost of this inventory holding is i; thus, the own rate of return on gold is D q - i, while the logarithm of the real stock of gold in terms of goods is n + q - /9. Demand takes the form n+
q -
p = v + l(Dq -
i) ,
(9)
where l and v are positive constants. Defining relative gold prices (q - p) as 0, asset market clearing may be written n+
O = v + l(DO -
r).
(9')
Hence, real capital gains depend on relative prices and the real interest rate. This asset-market relationship combines with the simple framework of the previous section consisting of an IS (Equation [1]), LM (Equation [2]), and the Phillips curve (Equation [3]). Consider the simple policy rule that revises the nominal shortterm rate according to the spread of the current over the steadystate relative commodity price, 0": D i = ¢(0 - 0 " ) .
(10)
The rule--incorporating gradualism and a concern for another ind i c a t o r - w o u l d suggest that higher relative commodity prices independently indicate that the real interest rate is lower than its steady-state value because investors want to hold more of the commodity asset. In the steady state, relative commodity prices are constant, with the long-run level of commodity prices equaling O* = v -
lr* -
n,
making the rule D i = ~(O -
v + l r * + n) .
Since both i and ~r are continuous, this equation, coupled with 586
C o n d u c t i n g M o n e t a r y Policy w i t h o u t a N o m i n a l A n c h o r
the Phillips curve, yields a real-rate-determination equation,
(10')
D r = d~(O - v + Ir* + n) - baz(r* - r) .
EOrl
[+'nv'+'+l+a'r+l
Equations (7') and (8) define a dynamic system in r and 0,
~---
DO
"4-
l/l
,
(n - v)/l
which determines two steady-state relationships between the real rate and relative commodity prices. Both of these schedules slope downward in (0, r) space and are drawn in the two panels of Figure 4. The upper panel examines the case in which the monetary policy parameter is small, 6 < baJl,
making the DO = 0 schedule flatter than the D r = 0 schedule. As the dynamic arrows suggest, this system is unstable so that no path leads to the steady state when r(0) differs from r*. Again, insufficient reaction by the central bank results in an explosive path because of the failure of i to adjust as quickly as "rr. If policy reacts more quickly, d~ > b a 2 / l ,
then the D r = 0 schedule is flatter than the DO = 0 schedule. As the dynamic arrows in the lower panel of Figure 4 suggest, for any given real short-term rate only one level of the market price of commodities sets off a dynamic path toward the steady state. That convergent path, known as the saddlepath, is labeled SS in the figure. While r is continuous, 0 is a market price that moves discretely when investors respond to the release of news. The existence of a saddlepath suggests that when an event makes r(0) differ from r*, financial markets can choose a unique level for commodity prices consistent with a stable return to the steady state. Along that path, the saddlepath, anticipated capital gains or losses satisfy the asset-market-clearing condition; fundamentals dictate the appropriate sequence of commodity prices. Along this transition path, the real rate differs from the nat587
Vincent Reinhart
@
I
o \
..................... - - - - - ~
DE
=
0
r-,
r-
r,
r
(9
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Figure 4. A Relative Commodity Price Rule ural rate so that inflation is changing. Just as with the excess-demand rule, the steady-state inflation rate depends on the length of that transition; a form of expression (A2) applies here, too. Any combination of low policy responsiveness (low ~b) and low demand sensitivity (low /) results in substantial long-lived consequences of a shock. Thus, while a relative commodity price rule can be stable, it does not determine a unique steady-state inflation rate; following only two of the three concerns of policy does not produce a satisfactory rule. 588
C o n d u c t i n g M o n e t a r y Policy w i t h o u t a N o m i n a l A n c h o r
Only the addition of a nominal anchor, a third concern, can tie down the system. The candidates for that role include the rates of goods or commodity price inflation or the rate of growth of the money stock. Any of those variables used in combination with a real commodity price target makes goods inflation in the steady state independent of initial conditions. For example, if the rule became Di = d~(0 - 0") + o~r,
then dynamics would be driven by a three-variable system in r, 7r, and 0, which is stable as long as d~ > a + ba2/l. Sufficient concern about the indicator complements concern about gradualism and inflation. Because the commodity serves as an indicator variable of no direct macroeconomic consequence, the steady state to this system is identical to the combination policy of responding to excess demand and inflation. That steady state depends on parameter values but not on the size of initiating disturbances. R e s p o n d i n g to t h e Y i e l d C u r v e
Unlike the asset just considered, we now examine the impact of a long-term asset in our model that has implications for the expenditure decision. The simplest device is to assume that the government introduces a claim that pays one unit of real purchasing power in perpetuity (as in Blanchard 1981). Let the long-term real return on this consol equal R. Expenditure, influenced by capital spending decisions with a long-time horizon, depends on the long real rate, y = al - a~R + e .
(1')
Changing expenditure behavior also changes the Wicksellian-inflation-determination equation; substituting (1') into the Phillips curve (3') results in D~r = ba2(R* - R) ;
(3")
the change in inflation now depends on the long-term real rate relative to the steady-state value, R*. Asset demands, however, depend on a shorter horizon. The expectations approach to the term structure hypothesizes that returns on comparable assets are equal over the same holding-period horizon. Maintaining this hypothesis, 589
Vincent
Reinhart i = R -
DR/$
(11)
+ ~r,
the instantaneous nominal return from a consol equals the shortterm rate of interest.~2 The arbitrage condition also takes the form, DR
= d~(R -
(11')
r),
showing that the market revises the long real rate according to the current slope of the real yield curve. Let us now consider a rule where the central bank takes advantage of the information incorporated in market pricing in setting policy: the change in the nominal short-term interest rate depends on an approximation to the slope of the nominal yield curve, x3 Di
= "y[(R + ~r) - i ] .
(12)
The similarities in (11') and (12) suggest that the policy maker is attempting to mimic the response of market rates. Control of the short-term real rate is given by the difference between the nominal short rate, determined by Equation (10), and inflation, determined by Equation (3"), D r = ~t(R -
r) -
ba~(R* -
R),
while the market revises the long-term real rate according to Equation (11), giving us two dynamic equations in the short- and longterm real rates. Figure 5 plots out the steady-state loci for this system. The D R = 0 schedule is a ray from the origin with unit slope because, by the form of the yield-curve relationship, short- and long-term real rates are equal in the long run. ~4 The D r = 0 schedule is flatter because the variation in the real short-term rate reflects partly policy decision, ~/(R - r), and partly market response to excess demand, b a 2 ( R * - R). At a point on the D r = 0 loci below R*, for example, the policy maker raises the nominal rate (because R > r) ~2This equation is an approximation to the duration-adjusted term structure (Mankiw 1986). ~3This is approximate because the true long-term nominal rate equals the longterm real rate plus long-term inflationary expectations, not ~r, the instantaneous inflation rate. More correctly, Equation (10) holds that policy makers respond to the slope of the real yield curve. ~4The presence of a liquidity premium shifts this schedule upward. 590
Conducting Monetary Policy without a Nominal Anchor DR
R
Q
=
0
Y
=
0
R*
Y
r
Figure 5. A Yield Curve Rule just as fast as the market raises inflationary expectations (because of the excess demand associated with R < R*). The intersection of the two schedules dictate the steady state, where r = R = R*. As the dynamic arrows in the figure suggest, this system responds cyclically to any shock that makes r different from R*. It can be shown that this cycle diverges from the steady state if ~/< @ and converges to the steady state if ~/ > t~. A large policy response, ensuring a convergent path to the steady state for any arbitrarily chosen (R, r) pair, presents a new problem. In the commodity-price case, a given short-term rate corresponded to a single commodity price that would lead the market to equilibrium. The joint assumptions that the market relied on fundamentals and that the system was stable implied one unique market price for commodities. No such argument can be made for the globally stable version of Figure 5. For any current r, any R that the financial market chooses generates a path that satisfies market fundamentals and stably fulfills expectations. Thus, a yield-curve rule does not tie down the market's initial choice for the long-term rate. Any arbitrary choice investors make will be confirmed by subsequent central bank action in setting shortterm rates. Ls Additionally, the steady state for nominal magnitudes tSThis uniqueness problem is associated with the work of Sargent and Wallace (1975) and McCallum (1986). 591
Vincent Reinhart depends on the initiating disturbance and the behavioral parameters. A policy stated in terms of relative yields places no special significance on the general level of yields. The long-run inflation rate depends on parameter values and the previous steady-state inflation rate. Adding a nominal anchor, or some reliance on a nominal magnitude, can render this rule determinate in two senses. First, a nominal anchor regains the uniqueness of the initial choice for the long-term rate. Stability will dictate only one possible long-term rate for each short-term rate. Second, a nominal anchor makes the steadystate inflation rate independent of initial conditions. Suppose, for instance, policy is written as
Di = ~/[(R + rr) - i] + et(-rr - ix).
(12')
The inclusion of inflation independent of the premium already appearing in nominal rates severs the automatic link between the market response, in terms of pricing R, and the policy response, in terms of setting r. It can be shown that the complete model determines real and nominal magnitudes and will be stable and unique for any nonzero ot as long as ~/ > t~; additionally, if not large enough to ensure stability by itself (~/ < t~), a nonzero ~/ lowers the value of a needed to stabilize the system.16 Thus, the presence of a nominal anchor renders the steady state unique while sufficient concern about the indicator stabilizes the system.
4. Conclusion The analysis of a simple macro model with forward-looking expectations suggests that a monetary authority can operate a stable and determinate gradualist interest-rate policy only as long as concern for inflation is teamed with monitoring other indicators, supporting the policy eclecticism of the recent Monetary Policy Report to Congress. In general, a policy rule cannot be based solely on relative prices or rates of return, including relative commodity prices and the slope of the yield curve, or respond solely to real magnitudes, such as the output gap.
IBIf policy follows a p u r e inflation rule, ~/ = 0, t h e n stability and u n i q u e n e s s requires that et > t~, that is, policy m u s t b e sufficiently c o m m i t t e d to t h e inflation target; a positive ~/ lowers this floor u n d e r a.
592
Conducting Monetary Policy without a Nominal Anchor First, a misapplied policy destabilizes the economy if it adjusts the short-term rate too sluggishly in response to a shock, regardless of the information variable guiding policy. This property, first noted by Wicksell (1898), lies at the root of academic mistrust of interest rate rules. Second, since these indicators already build in current inflationary expectations, a policy responding only to them, even if stable, lacks direction to force the economy to a particular long-run inflation rate. More specifically, the steady state depends on the transition path, so that any shock becomes incorporated into the long-run inflation rate. Third, a rule conditioned on a forward-looking asset price may ratify any potential price that the financial market chooses~ making it impossible to determine a unique equilibrium path. A yield-curve rule, in particular, has this uniqueness problem that implies that policy could set off on any of a number of paths by simply following the market's initial arbitrary choice. Any of these indicator variables, however, complement the use of an inflation target. Received: October 1989 Final version: October 1990
References Angell, Wayne D. "'Commodity Prices and Monetary Policy: Empirical and Theoretical Considerations." Paper presented at the annual meeting of the Virginia Association of Economists, March 16, 1989. Barsky, Robert B. "'The Fisher Hypothesis and the Forecastability and Persistence of Inflation.'" Journal of Monetary Economics 19 (January 1987): 3-24. Barsky, Robert B., and Lawrence H. Summers. "'Gibson's Paradox and the Gold Standard.'" Journal of Political Economy 96 (June 1988): 528-59. Begg, David K.H. The Rational Expectations Revolution in Economics: Theories and Evidence. Baltimore: Johns Hopkins Univ. Press, 1982. Blanchard, Olivier J. "Output, the Stock Market, and Interest Rates." American Economic Review 71 (March 1981): 132-43. Boughton, James D., and William H. Branson. "'Commodity Prices as Leading Indicators of Inflation." Working Paper. International Monetary Fund, October 1988. Boyce, William E., and Richard C. DiPrima. Elementary Differ593
Vincent Reinhart ential Equations and Boundary Value Problems. New York: John Wiley and Sons, 1977. Branson, William H., and Dale W. Henderson. "The Specification and Influence of Asset Markets." In Handbook of International Economics, edited by Ronald W. Jones and Peter B. Kenen, 749805. New York: North-Holland, 1985. Dornbusch, Rudiger. "Expectations and Exchange Rate Dynamics." Journal of Political Economy 84 (December 1976): 1161-76. Federal Reserve. "'Monetary Policy Report to Congress.'" Federal Reserve Bulletin 76 (March 1990): 107-19. Fuhrer, Jeffrey, and George Moore. "'The Natural Rate of Interest and Monetary Policy." Working Paper. Board of Governors of the Federal Reserve System, March 1989. Gagnon, Joseph, and Dale W. Henderson. "'Nominal Interest Rate Pegging Under Alternative Expectations Hypotheses." Working Paper. Board of Governors of the Federal Reserve System, May 1988. Hahn, Frank H. "Monetarism and Economic Theory." Economica 47 (February 1980): 1-17. Hallman, Jeffrey J., Richard D. Porter, and David H. Small. "'M2 per Unit of Potential GNP as an Anchor for the Price Level." Board of Governors of the Federal Reserve System Staff Studies 157, 1989. Johnson, Manuel H. "Issues Facing Monetary Policy in 1989." Speech before the National Association of Business Economists, Washington, D.C., February 28, 1989. Mankiw, Gregory N. "The Term Structure of Interest Rates Revisited." Brookings Papers on Economic Activity 17, no. 1 (1986): 61-96. McCallum, Bennett T. "Some Issues Concerning Interest Rate Pegging, Price Level Determinacy, and the Real Bills Doctrine." Journal of Monetary Economics 17 (January 1986): 135-60. Meek, Paul. ed. Central Bank Views on Monetary Targeting. New York: Federal Reserve Bank of New York, 1982a. Meek, Paul. U.S. Monetary Policy and Financial Markets. New York: Federal Reserve Bank of New York, 1982b. Reinhart, Vincent. "Interest Rate Smoothing and Staggered Contracting." Journal of Economics and Business 42 (February 1990): 1-16. Rodriguez, Carlos A. "'A Simple Keynesian Model of Inflation and U n e m p l o y m e n t under Rational Expectations." Weltwirtschaftliches Archiv 114, no. 1 (1978): 1-11. 594
Conducting Monetary Policy without a Nominal Anchor Sargent, Thomas J., and Neil A. Wallace. "" 'Rational' Expectations, the Optimal Monetary Instrument, and the Optimal Money Supply Rule." Journal of Political Economy 83 (April 1975): 241-54. Sheffrin, Steven M. Rational Expectations. Cambridge: Cambridge University Press, 1983. Whitesell, William C. "Inflation-Fighting with Interest Rate Rules." Working Paper. Board of Governors of the Federal Reserve System, March 1990. Wicksell, Knut. "'The Influence of the Rate of Interest on Commodity Prices." Lecture to the Economic Association in Stockholm, 1898. In Selected Papers on Economic Theory. Cambridge: Harvard University Press, 1958, 67-89.
Appendix A: The Long-Run Consequences of an Output-Gap Rule As explained in Section 2, a policy rule responding only to excess demand reduces to a single linear differential equation in the short-term real interest rate, Equation (8), the solution to which takes the form
r(t) - r* = Jr(0) - r * ] . e x p [ - ( ~ - b)aut]. When the rule is stable (~ > b), the impact of a shock that makes the current short-term rate, r(O), differ from the natural rate, r*, decays exponentially. During that adjustment, however, the inflation rate varies in response to the difference in expenditure from potential. Indeed, the change in the inflation rate up to period t consists of the cumulative impact of excess demand, t
•r(t) = at(0) + b f0 [y(s) - k i d s , or, remembering that excess demand is written as the difference between the real and natural rates, ~(t) = ~(0) - ba2[r(O) - r*]
exp[-(8 - b)a2s]ds.
Performing this integration, rr(t) = ,r(0) + [r(0) - r*]b. {exp[-(8 - b)a2t] - 1}/(8 - b ) .
(A1) 595
Vincent Reinhart N o t e t h a t as t ---> ~, ~r(t) = rr(0) - [r(0) - r*]" thus, initial c o n d i t i o n s , rr(0) a n d state inflation.
b/(8 - b) ;
(A2)
r(O), figure i m p o r t a n t l y in s t e a d y -
Appendix B: List of Variables y = k = i = ~r = r = m = /9 = p* = n q 0 R
596
= = = =
rea] i n c o m e ; potential output; n o m i n a l s h o r t - t e r m i n t e r e s t rate; e x p e c t e d inflation rate; e x p e c t e d real s h o r t - t e r m i n t e r e s t rate; n o m i n a l m o n e y stock; g o o d s - p r i c e index; p r i c e level t h a t clears t h e m o n e y m a r k e t at its s t e a d y - s t a t e values g i v e n t h e c u r r e n t n o m i n a l m o n e y stock; stock o f t h e s t o r a b l e c o m m o d i t y ; nominal price of the storable commodity; relative c o m m o d i t y prices; l o n g - t e r m real r e t u r n on t h e consol.